1. Field of the Invention
This invention relates to a defuzzifying apparatus for realizing an ambiguous-information-based or ambiguous-knowledge-based fuzzy inference such as flow of control, pattern recognition, decision making, etc., by hardware.
2. Description of the Related Art
In recent years, flow of control and pattern recognition by fuzzy inference have been widely used. Generally an inference is done by a "if XXX then YYY" construction, namely, the "if-then" rule. In a fuzzy inference, either of the above XXX and YYY portions is constructed as a fuzzy set. The above portion "if XXX" is called the antecedent, and the above portion "then YYY" is called the consequent.
The algorithm of fuzzy inference consists of the following four processes:
(1) Decide to what extent the given input and the antecedent match each other for every inference rule.
(2) Obtain the result of inference for every inference rule from the consequent according to the result of the process (1).
(3) Integrate a number of inference results.
(4) Obtain a definite output value, as the final output, from the integrated inference result. This process is called a "defuzzification".
Various operations such as minimal value operations, maximal value operations, algebraic product operations and algebraic sum operations have been proposed to accomplish the above processes (1), (2) and (3). Regarding the defuzzifying process (4), however, it is a common practice to use the so-called center of gravity method (CG method) in which a definite output value is obtained from a center of gravity of the inference result expressed by a fuzzy set on a support set for the consequent.
FIG. 9 of the accompanying drawings shows the general principle of the fuzzy inference algorithm by a minimum-maximum-gravity method which is very popular in the art. In this method, minimal value operations and maximal value operations are used to perform the processes (2) and (3), respectively. Further the above-mentioned CG method is used, as the defuzzifying process of the process (4), to calculate a definite output.
The practical operation for calculating the center of gravity is expressed by tile following equation: ##EQU1## wherein, as shown in FIG. 10, k is the number of divisions in the support set for the consequents, i is the address of each division, .mu..sub.1 is a degree of grading representing the inference result of address i, and w.sub.1 is a weighting factor representing the position of address 1 on the x coordinate.
Recently it has been confirmed that there would be no inconvenience with the inference result even if the YYY portion of the consequents is a constant (singleton) rather than a fuzzy set. This method is called "the simplified method"; since the number of calculation processes is considerably reduced, the number of practical applications for this method in various fields, especially in control technology is on the increase. This is called "simplified fuzzy inference".
The algorithm of a simplified fuzzy inference will accordingly be described in conjunction with FIG. 11. A given input x is compared and collated with a membership function described in the antecedent for each rule 1, 2, 3, to calculate a value corresponding to the input. This value is called "the degree of matching"; the inference result for each and every rule can be obtained by cutting off the consequent by tile degree of matching.
In a simplified fuzzy inference, since the consequent consists of a singleton, the degree of matching of the antecedent is regarded as the degree of grading of the inference result.
Then the degrees of grading of inference results obtained from a number of rules must be integrated to output only a single definite value. This process is called "the defuzzification", in which, as with the conventional art, a center of gravity is obtained.
This calculation is expressed by the following equation: ##EQU2## where 1 is each singleton constituting the consequents, n is the sum of singletons, w.sub.1 is the position of each singleton on the x coordinate, i.e. a constant, .mu..sub.1 is a degree of inference grading corresponding to each singleton. The numerator will be a weighted sum value of individual inference grades, while the denominator will be a simple sum value. When the weighted sum value is divided by the simple sum value, a center of gravity will be obtained as a definite output of fuzzy inference.
One of the features of the gravitational method as a defuzzifying process is that, as shown in FIGS. 12a, 12b and 12c, the center of gravity will not fluctuate even if the degree of grading of the inference result is varied. Either if the degree of grading of the inference result is large as shown in FIG. 12a or if it is small as shown in FIGS. 12b and 12c, their respective center of gravity are identical unless they are not 0. Consequently, tile following problems were encountered.
(1) Assuming that the defuzzifying process is realized by hardware, an abnormal definite output value will be outputted only in the presence of a small amount of noise or error in the inference result. For example, a center of gravity due to noise will be outputted as correct even in the absence of any grade of inference result as shown in FIG. 13a.
(2) In the conventional method, if an input does not match any of the inference rules at all, the degree of grading of the inference result is 0. At that time, either of the numerator and denominator of equation (1) will be 0 so that the center of gravity cannot be calculated; therefore, assuming that the defuzzifying process is realized by hardware, an abnormal value will be outputted if the input does not match any of the inference rules at all.